No you will not qualify with "severe pain". Migraines are bad, yes.. but not the "severe pain" they're talking about. But yes, it depends on the doctor you talk to.
Unless you're actually diagnosed with a cause for the migraines, you will have a very hard time getting anywhere in terms of getting a medicinal card. MS, epilepsy, glaucoma, Hep. C., things like that..

Usually it needs to be "chronic" pain (mind the pun) in order to get considered for a medicinal card.

Bob.